We summarize the usual Lie group theoretical symmetry picture
for quantum and classical dynamics in Table 1, with the quantum
case illustrated in matching correspondence of the noncommutative
coordinate and (commutative) complex number coordinate results.
Each one-parameter group of unitary transformation generated by
a Hermitian operator on the Hilbert space is a Hamiltonian flow
and the preserved Poisson bracket among functions on the
latter is the exact correspondence of the operator Poisson
bracket given by \(\frac{1}{i(\hbar )}\)  times the commutator.
Each observable \(\hat{\beta }\)  corresponds to the expectation
value function \(f_{\!\scriptscriptstyle \hat{\beta }}\)  collection of which, with
the noncommutative product \(\star _{\!\scriptscriptstyle \kappa }\)  is an
exact copy of the observable algebra itself. However, a
generic Hamiltonian function of \(f(z_n,\bar{z}_n)\)  or \(f(\psi )\) 
can be matched to an observable as an operator only if the
flow generated preserves the full Kaḧler structure [1]}.
Natural sets of complex coordinates are obtained as the set of
expansion coefficients in terms of sets of orthonormal basis for
the Hilbert space. Such a set of complex coordinates are canonical
in the sense that the real and imaginary parts give exactly pairs
of canonical real coordinates. Some of the notation and results
may be easier to appreciate through checking Appendix.
The Lie group symmetries can be seen, as illustrated, in the
noncommutative geometric point of view with the position
and momentum observables taken as phase space coordinates.
The phase space and the observable algebra as functions on it can
be obtained from an irreducible representation of the relativity
symmetry in the quantum case. A general symmetry generator,
however, can be a generic Hamiltonian function as an observable,
outside the Lie algebra of the relativity symmetry. The
noncommutative geometric point of view provides also
the natural setting to look at the quantum reference frame
transformations we are interested in here. An important
point to note though is that, unlike the Lie group symmetries,
the quantum reference frame transformations for a system
cannot be considered on its phase space alone. As we have seen
above, the plausibility of nontrivial entanglement between
the system and the old or new frame of reference as another
physical system demands a more complicated formulation.
It is still interesting to compare the new kind of symmetry
transformations with the old ones.
